I was afraid you'd ask that. I don't know of such an
application. You would more than likely do it on the server rather
than the client. Check places like Sorenson, or Flex Coders or some
of the other Flash sites.

This thread's a few months old but I thought I'd mention that
I have developed a Flex component for loading and playing animated
gif files. Now there's no need to hack together some weird IFrame
solution, or figure out how to do the server-side conversion of all
your GIFs. You can try the demo version for free and the commercial
version is $50.

Here's the description I posted on Flex Exchange:

The AnimatedGifLoader loads and plays animated GIF files.
This component adds support for animated GIFs, a feature that has
been lacking from Flex.

This works just like the SWFLoader component from the Flex
Framework. Use it inline in your MXML code or by using
Actionscript. This component supports everything the SWFLoader
component does, with a few additions.

No, unfortunately this component does not enable animated
GIFs within Text components that support htmlText. The root reason
for this is that the htmlText property is part of the
flash.text.TextField class, and the classes in any of the flash.*
packages are not available. So to enable animated gifs in a
TextField or TextArea, I'd have to be able to see how
flash.text.TextField handles the <img> tag.

After taking an initial look, I don't think I'm going to be
able to get a new component based off TextArea to support animated
gifs. I'll let you know if I come up with anything, but I wouldn't
hold my breath. Modifying anything that uses the htmlText property
would require access to classes that are built into the flash
player, instead of classes available in the flex framework. So
while it's probably not impossible to figure out a solution, it's
going to pretty difficult.